快速扩散方程的超快速消失解
Super fast vanishing solutions of the fast diffusion equation
摘要：
我们将推广B.Choi、P.Daskalopoulos和J.King的最近结果。对于任何$n\ge 3$、$0<m<\frac{n-2}{n+2}$和$γ>0$，我们将构造快速扩散方程$u_t=\frac{n-1}{m}Δu^m$在$\mathbb{R}^n\times (t_0,T)$中的子解和上解，$t_0<T$，它们以$(T-t)^{\frac{1+γ}{1-m}}$的速率衰减为$t\nearrow T$。从而得到了柯西问题$u_t=\frac{n-1}{m}Δu^m$在$\mathbb{R}^n\times (t_0,T)$中，$u(x,t_0)=u_0(x)$在$\mathbb{R}^n$中的唯一解的存在性，当$u_0$满足适当的衰减条件时，解以$(T-t)^{\frac{1+γ}{1-m}}$的速率衰减为$t\nearrow T$.
英文摘要：
We will extend a recent result of B.Choi, P.Daskalopoulos and J.King. For any $n\ge 3$, $0<m<\frac{n-2}{n+2}$ and $γ>0$, we will construct subsolutions and supersolutions of the fast diffusion equation $u_t=\frac{n-1}{m}Δu^m$ in $\mathbb{R}^n\times (t_0,T)$, $t_0<T$, which decay at the rate $(T-t)^{\frac{1+γ}{1-m}}$ as $t\nearrow T$. As a consequence we obtain the existence of unique solution of the Cauchy problem $u_t=\frac{n-1}{m}Δu^m$ in $\mathbb{R}^n\times (t_0,T)$, $u(x,t_0)=u_0(x)$ in $\mathbb{R}^n$, which decay at the rate $(T-t)^{\frac{1+γ}{1-m}}$ as $t\nearrow T$ when $u_0$ satisfies appropriate decay condition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
